I N S T I T U T D ' A E R O N O M I E S P A T I A L E D E 3, avenue Circulaire, UCCLE - BRUXELLES 18
B E L G I Q U E
A E R O N O M I C A A C T A
A - N° 46 - 1966
Collision frequencies and energy transfer : ions by P. BANKS
B E L G I S C H I N S T I T U U T V O O R R U I M T E - A E R O N O M I E 3, Ringlaan, UKKEL - BRÜSSEL 18
FOREWORD
"Collision Frequencies and Energy Transfer % Ions" is the second of a series of two papers dealing with the general subject of elastic collision processes and energy transfer applied to the con- stituents of the upper atmosphere. This series will be published in Planetary and Space Science during 1966.
A.VANT-PROPOS
"Collision Frequencies and Energy Transfer : Ions" est le second de deux travaux dont le sujet général est l'étude des processus de collision élastique et de transfert d'énergie appliqués aux con-
stituants de l'atmosphère supérieure. Les résultats de ces recher- ches seront publiés en 1966 dans la revue Planetary and Space Science.
VOORWOORD
"Collision Frequencies and Energy Transfer s Ions" is het tweede in een reeks van twee werken die handelen over het algemeen onderwerp ; processen van elastische botsing en energie-overdracht, toegepast op de bestanddelen van de hogere atmosfeer. Deze reeks zal gepubliceerd worden in Planetary and Space Science in de loop van 1966.
VORWORT
"Collision Frequencies and En®rgy Transfer : Ions" ist die zweite Arbeit einer Serie von zwei Abhandlungen, die sich auf das ge- neral Problem der elastischen Stossprozesse und der Energieübertragung
in der höheren Atmosphäre bezieht. Diese Arbeit wird im laufenden 1966 in Planetary and Space Science herausgegeben werden.
COLLISION FREQUENCIES ÄND ENERGY TRANSFER : IONS by
Peter BANKS
Abstract
An investigation has been made of ion collision frequen- cies and energy transfer for conditions of gaseous thermal nonequi-
librium. Following the methods of a development devoted to electron collisions, the problems of ion-neutral and ion-ion elastic colli- sions have been considered under the assumption of separate Maxwel- lian velocity distributions. Expressions for the nonequilibrium mo- mentum transfer cross sections, collision frequencies, and energy
transfer rates are given for neutral gases on the basis of a polari- zation potential and for ions on the basis of the Coulomb interaction.
Next, the problem of ion-neutral resonance charge exchange is intro- duced in its effect upon collision cross sections and ion energy
transfer rates. An analysis is made of gathered laboratory data to arrive at suitable expressions for the Resonance charge exchange and momentum transfer cross sections for H , 0+, He+, N2 +, and 02 +. Using these results, resonance charge exchange and momentum transfer collision frequencies are derived. The resonance ion-neutral energy transfer rates are then presented and compared with expressions used in previous studies of ion temperatures.
Résumé
Nous étudions les fréquences de collision et le transfert d'éner- gie des ions dans des conditions de-gaz en non-équilibre thermique. Sui- vant les méthodes d'une recherche consacrée aux collisions des électrons, nous considérons les problèmes des collisions élastiques ion-particule neutre et ion-ion, sous l'hypothèse de distributions de vitesses maxwellien- nes différentes. Les expressions des sections efficaces de transfert de moment en non-équilibre, les fréquences de collisions et les taux de trans- fort d°énergic sont donnés pour lcogaa neutreosur la base d'un potentiel de polarisation et .pour les ions sur la base d'une interaction coulombienne.
Ensuite, nous considérons le problème de l'échange dechaage «atuet ipnrèfc pàrticule neutre, du point de vue de ses effets sur les sections efficaces de colli-
sion et sur les taux de transfert d'énergie. Nous analysons ensuite des données de laboratoire pour en tirer des expressions appropriées pour les sections efficaces d'échange de charge par résonance et de transfert d'éner- gie pour H4", Ô+, He+, N2 + et? 02 +. Les fréquences de collision pour l'échan- ge de charge par resonance et pour le transfert de moment sont déduits de ces résultats. Nous présentons alors les taux de transfert d'énergie par réso- nance ion-particule neutre et nous les comparons avec des expressions utilisées
d à a è & l e s ' i ë t u â e s r a n t é r l f e t t r f e s ' i o n i q u e s .
Samenvatting
In deze tekst worden de botsingsfrekwenties en de energie- overdracht bestudeerd voor ionen van gassen welke niet in thermisch
evenwicht zijn. De elastische botsingen tussen ionen en neutrale deeltjes evenals tussen ionen onderling worden bestudeerd volgens de methode welke uiteengezet werd voor de botsingen die de electronen ondergaan in de on- derstelling van een verschillende maxwelliaanse snelheidsverdeling. Er worden uitdrukkingen gegeven voor de werkzame doorsnede voor de over- dracht van hoeveelheid van beweging ingeval de gassen niet in thermisch evenwicht zijn» evenals voor de botsingsfrekwentie en de mate van energie- overdracht door de ionen. Er wordt een studie gemaakt der gegevens die in verscheidene laboratoria gevonden werden ten einde geschikte uitdruk- kingen te bekomen voor de ladingsuitwisseling bij resonantie en de werkzame doorsneden voor de overdracht van hoeveelheid van beweging in-
geval van H*, 0+, He"*V KL+ en 0 * . Gebruik makende van deze resultaten werd' de ladingsuitwisseling en botsingsfrekwentie voor de overdracht van hoeveelheid van beweging.bepaald. Des -mate van energieoverdracht bij ladingsuitwisseling in resonantietoestand tussen ionen en neutrale deeltjes wordt dan berekend en vergeleken met de uitdrukkingen gebruikt in vroegere studies betreffende ionentemperaturen*
Zu sammenfassung
Eine Abhandlung der lonenzusanpenstossfrequenz und der Energie- übertragung für nicht thermischem Equilibrium wird ausgeführt. Nach einer Methodedie für Elektronen entwickelt wurde, wird das Problem der Zusam- menstösse zwischen Ionen und ungeladenen Teilchen sowie zwischen Ionen im
Falle verschiedenen Maxwell - Verteilungsfunktionen analysiert» Aus- drücke für Momentumstreuquerschnitten, Zusammenenstossfrequenzen und Energieübertragungskoeffizienten werden mit Hilfe einer Polarisations- potentiales für ungeladenen Teilchen und mit Hilfe einer Coulomb-Inter- aktion gegeben» Dann betrachten wir das Problem der Resonanzum-
ladung zwischen Ionen und ungeladenen Teilchen hinsichts des Einflusses auf die Streuquerschnitte und auf die Energieübertragungskoeffizienten„
Die Experimentalergebnisse werden analysiert, um Ausdrücke der Umladungs- querschnitte und der Energiettbertragungskoeffizienten für H4", 0+, He+, N2 +
und 0 + zu erreichen» Daraus folgen Ausdrücke der Zusansmenstossfrequenze für Umladung und Momentumübertragung. Endlich werden die Energiettber- tragungskoeff izienten für Ionen und ungeladenen Teilchen vorgestellt und mann vergleicht diese' Ausdrücke mit anderen Formeln, die im Problem der lonentemperatur gebraucht wurden.
I„- INTRODUCTION
In order to discuss the thermal budget between electrons and ions and neutral gas particles in the ionosphere it is necessary to know all of the energy exchange rates between the different
constituents, taking into account their specific temperature de- pendences under conditions of thermal nonequilibrium. The problem of energy transfer from an electron gas with a Maxwellian velocity distribution as a result of elastic collisions has been discussed in a previous paper (Banks, 1966) using an equation which is of ge-
neral applicability to problems of elastic energy transfer. In this paper the study of energy tiansfer is continued with emphasis being placed upon the atmospheric ions. Thus, in Section II we enter into a dis- cussion of the collision frequency and energy transfer equations which are valid for elastic ion-neutral and ion-ion energy transfer.
In contrast to the electron gas problem, there exists for ions a second method of energy loss by means of resonance charge exchange which, while fundamentally elastic in nature, cannot be
described by the equation of collisional energy transfer. The problem of resonance charge exchange between ions and their parent neutral gas is discussed in Section III with respect to both collision fre- quencies and ion energy processes„ Following an analysis of labo- ratory and theoretical cross section data numerical results for the resonance ion-neutral energy transfer rates are obtained and compared with expressions which have been used in different theoretical analyses of ion temperatures.
4 . -
II.- N O N R E S O N A N T ION COLLISIONS
The equations describing collisional energy transfer have been discussed previously (Banks, 1966) for the example of two gases with Maxwellian velocity distributions characterized b y the tempera-
tures T ^ and T^ and particle m a s s e s m ^ and m ^ . T h e energy transfer rate, dU^/dt, from the first gas is given by
dU. m m _
~ " " 3 nl 7 - ^ 2 ( k Tl " k T2 > V1 2 ( 1>
dt (n^-Ki^)
where n ^ is the first gas number density, k is Boltzmann's constant, and v^^ is the average collision frequency of a single particle of the first type in the second gas. Since the density, masses and • temperatures are independent of the interaction forces between the different particles it is seen that it is the collision frequency which must be determined in order for the energy transfer rate between
arbitrary types of gases to be k n o w n . The result is (see Banks, 1966)
4 / 8k \ 1/2 / I T \ 1/2 _
w h e r e n^ is the second gas number density and Q ^ is the average m o m e n t u m transfer cross section, given in terms of the velocity dependent m o m e n t u m transfer cross section5 qD» and the interparticle relative v e l o c i t y , g,as
QD = K3j f g5 qD exp ( - K g2) dg (3)
r ! ^ L m.
2kT 2kT n K = I +
1 m2
With equations (1), (2), and (3) it is possible to derive accurate expressions for both energy transfer rates and collision frequencies for gases of arbitrary temperature and particle mass once the velocity dependent momentum transfer cross section is known. Thus, the problem of ion elastic energy loss revolves, essentially, about the choice of proper cross sections needed to describe the interaction be-
tween the ion gas and the gas with which it is mixed. In the following analysis we will be concerned with two particular situations which are of some practical importance. The first involves the derivation of energy transfer rates and collision frequencies for ion-neutral gas mixtures under conditions where there are only elastic collisions.
Next, we consider the problem of ion-ion gas mixtures when each ion species has its own separate Maxwellian velocity distribution.
1. Ion-Neutral Collisions
At low temperatures the most important ion-r.eutral inter- action arises from an induced dipole attraction which has a poten-
2 4
tial q p = - ae /2r . Here, a is the neutral atom polarizability, e is the electric charge, and r is the radial separation. This poten- tial corresponds to the force law acting between Maxwellian molecules and the details of the transport coefficients under conditions of ther- mal equilibrium are well known. In particular, it is found the collision -1/2 cross section is proportional to T and that the collision frequency is independent of the temperature.
For temperatures greater than 300°K, the induced dipole force of attraction is countered by a short range quantum mechanical repul- sion. For sufficiently high temperatures, the polarization contribution is negligible and the collision cross section becomes nearly constant, yielding a collision frequency which varies as T * ^2. Unlike the polari- zation force, which is independent of the chemical nature of a given ion and depends only upon the atomic polarizability of the neutral gas,
the nature of the repulsive force is linked directly to the details of the ion and neutral orbital electron structure. Thus, variations in cross section are to be expected for different ions in the same neutral gas at elevated temperatures. '
Because there exist virtually no data for either collision frequencies or cross sections at high temperatures, the study of ion- neutral collisions must be based upon the assumption that the pola- rization force is of dominating importance. A detailed evaluation of the ion-neutral interaction potential has been made by Dalgarno et al.
(1958) who terminated the polarization potential at a small atomic radius and added an elastic sphere repulsive potential to simulate the quantum repulsion effets. This theoretical cross section was then compared with reported measurements of ionic mobilities. For equal ion and neutral temperatures below 300°K the theoretical results appear to be accurate. Above 300°K, the detailed variation of the momentum transfer cross section depends upon the specific ion and neutral gas.
With there being only a few experimental data for tempera- tures above 500°K, it is necessary to accept the collision equations developed for the polarization force alone as representing the true interaction over the entire range of ion and neutral temperatures discussed here. Accordingly, we take the ion-neutral velocity de- pendent momentum transfer cross section for singly charged ions to be, following Dalgarno, et al. (1958),
, 2 1/2
qD - 2.21 ) c m , (4)
where a is the neutral gas atomic polarizability, |i is the ion-neutral reduced mass, e is the electron charge, and g is the ion-neutral par- ticle relative velocity.
Using equation (3) the average momentum transfer cross section for ion-neutral collisions becomes
3V"2 . . . / 4.88ae s
"d- S » C-iT—)
2 1/2
rkT.- lcT -i
+ _ i l m . m
l n -1/2
(5)
o r , numerically,
Qd = 13.3 x 10 • W V 1 / 2
l
k
r
T„ T -i i + - n L A . A Jl n
•1/2
cm (6)
where A ^ and A ^ are the particle masses in atomic mass units (a.m.u.),
|i is the reduced mass in a.m.u,, a is the atomic polarizability in A -24 3 °
units of 10 cm , and the subscripts apply to ions and neutrals, respectively.
For conditions of equal temperatures the mass factors of equation (6)cancel and w e have
3V"2
V V V - I T n3'2 ( 4.88ae
kT n
1/2
(7)
which agrees with the value used by Dalgarno, et a l . (1958) in their calculations of ion mobility.
Using equations (5) and (2) the average ion-neutral collision frequency for energy transfer is obtained as
_ / ae v = 2.2In n ( —
in n \ (i
2 1/2
(8) or.
" , a ,n-9vi n = 2 . 6 x 1 0 n ^ - ) sec ,
Co\
1 / 2 "IA
(9)
independent of either gas temperature. Actually, as shown by Morse (1963), although we have derived the present expression for the average momentum transfer cross section on the basis of separate Maxwellian velocity distributions, for the case of Maxwellian molecules the cross section, collision frequency, and energy transfer raté are independent of the actual velocity distribution used.
Table 3=7, taken from Hasted (1964) and Dalgarno(1961a), gives values of the atomic polarizability. for the atmospheric gases.
Table 1 Polarizabilities of Neutral Gases -24 3 Neutral gas a (10 cm )
o
N2 1.76
02 1.98
H2 0.81 0 0.77 H 0.67 He 0.21 N 1.13
Equation (8) can be compared with an ion-neutral collision frequency which follows from the work of Chapman and Cowling (1952).
In the diffusion of ions across a magnetic field, a collision inter- val -s appears which can be used to define a collision frequency v by the relation v » l/ic. The value of t is related to the ion- neutral diffusion coefficient which, in turn, can be calculated directly for the polarization potential. This process yields an ion-neutral collision frequency, using the cross section of Dalgarno, c£ al. (1958), given by
n„m„ + n m , .„
™ i i n n „ / 2 .1/2
v. = — — — — — 2.21n (a e n) (10)
in m.m ' N /
l n
9 . -
where n. is the ion density and n^ is the neutral gas number density.
Comparing this result with equation (8) for the energy transfer colli- sion frequency, it is seen that equality follows only if n^ = n^.
For other der.sity conditions, in particular where n , « n^, the "two collision frequencies are not the same. Dalgarno (1961b) has given ion-neutral collision frequencies based upon equation (10) under the assumption that n^ < < n ( n ) . For this approximation it follows that his results are smaller than those presented here by the ratio n/m .
For the example of 0 in N^ this corresponds to a factor of 0.64. + i
Using the ion-neutral collision frequency given by equation (8), the rate of ion energy transfer in a neutral gas is found from equation (1) to be
dU 2.21 n ,
7 Tdt i n (m. + m ) = " 3 n*n„ (m j. m ^ . ( a n e V k(T. - T ) (11) i n or.
l n
1/2
x , _ A o _ . -J -i.
— — = - 6;8 x 10 n.n — , . . (T.- T ) ev cm sec
dt i n ( A „ + A ) v i n
1 n (12)
Appropriate values for the neutral gas polarizabilities are given in Table 1 and have been used to obtain the energy transfer rates listed in Table 2. As a consequence of the particular force law for ion-neutral collisions, the energy transfer Urates depend only upon the difference in ion and neutral gas temperatures. In general there is a variation by a factor of 3.6 between the different
loss rates considered. Further, except for the collisions H * - He"*" and He- H, the differences in the rates are determined primarily by the varying
mass factors rather than the changes in atomic polarizability.
1 0 . -
It is also noted that these ion energy loss rates are considerably larger than those found (see Banks, 1966) for elastic electron-neutral collisions. As an example we may consider the 0 * - N rate as, omit-
-14 3 -1 ting common density and temperature factors, 6.6 x 10 ev cm °K sec
— 1 fi o j while the e-N2 rate is, at Tg = 1000°K, only 1.55 x 10 ev cm °K sec This difference of a factor of 430 arises from the enhanced ion-
neutral cross section and advantageous mass factors in equation (1), mitigated, of course, by the much more rapid average electron velo- city.
Table 2 Elastic Ion-Neutral Energy Transfer Rates
3 -1 -1 Collision Pair -dl^/dt (ev cm sec °K )
n.n (T -T ) l n i n 0 - N-
+
20 - 0„
+ +
2 0 - He 0 - H+
H - 0
+
H - N-
+ +
2 H - He He - 0+
He - H
+
He - N,
+
6.6 x 10 -14
6.5 x 10"1 4 2.8 x 10 -14 3.3.x 10 -14
3.5 x 10~1 4 3.1 x 10"1 4 5.5 ax 10 -14 5.4 x 10 -14
-13 1.0 x 10 5.3 x 10"1 4
The results given here can be compared with values which have been used in previous studies of ion temperatures in the upper atmosphere. In a recent paper Willmore (1964) has adopted values for the helium and oxygen ion rates the values 3.3 x 10 and 3.6 x lo"*4
I r -
respectively. In comparing these rates with those listed in Table 2 . it appears that Willmore's helium ion cooling rate, which is based upon the diffusion collision frequencies of Dalgarno (1961b), is too
low by a factor of 0.63 (see equation 10 and discussion). Likewise, the oxygen ion energy transfer rate is too large by a factor of 1.3.
2 o Ion-Ion Collisions
The average momentum transfer cross section for two separate ion gases, each having its own Maxwellian velocity distribution is, Banks (1S66) ,
2 2 _ re (Z^Z^e ) In A
= 7 •
D k Tx kT2 ~|2
L ml + m2 T
(13)
where Z. - are the respective ion charges and the parameter A is given
•L»^
by
2 € A
Z1Z26
(14)
Here € is the average energy of relative motion between ions and the plasma Debye length, is given by
4n e V i + V i - kT, kT,
1 2 ~
Equation (13) can be evaluated numerically to give
(15)
4.4 x 10 In A
T1 V l 2 L ^ + v
cm . 2 (16)
using the previous notation.
1 2 . -
The average ion-ion energy transfer collision frequency i s , from equation ( 3 ) ,
or.
12 3
(Z Z e2)2 In A 4 lfs— 1 2
V2n n
r
kT kT -| — +—
3 / 2 (17)-2 ( Z1Z2) l n A V12 = X 1 0 n2 T "13/2
_ A, A„
J
cm (18)
1 " 2
To find the average collision frequency of a thermal ion in its own gas we let = T = T. and m, - m. = m . . Thus,
1 2 i • 1 2 l
v — n — 12 i 3
2 4 . n Z. e In A
i
(kT.) 3 / 2 (19)
which is a factor of 4/3 larger than the ion collision frequency given by Chapman (1961) for total scattering effects.
The total rate of energy transfer between two ion species?having Maxwellian velocity distributions iis obtained from equation (1) by using the average ion-ion energy transfer collision frequency given by
equation ( 1 7 ) . Thus,
dt
| r— .
|/2n nx n 2 e k(Tx- T2) ln A r k Ti k T 2 i 1
m, m„ + — ~ 2 1 2 L i j m2 J
(20)
or,
1 3 . -
dU1 2.18 x 10"5 nx n (Tj- T2> In A _3 _1
- — = - r ev cm sec dt r- T, -i 3
A1A2
A1 A2 J
2 (21)
which agrees with the result Longmire (1963). Thus, the concept of binary collisions permits the exact description of the ion-ion energy transfer rate under the condition that each species has its own
Maxwellian velocity distribution.
At this point it is now possible to make a comparison of the elastic and resonant ion-neutral energy transfer rates with the calculated ion-ion values. If we disregard the density and temperature differences it is found from Tables 2 and 11 and equation (21) that the ion-ion rates are generally larger by at least a factor of 10 for ion temperatures near 1000°K. This would imply that the ion gases are more strongly coupled to each other than to the neutral gases. However, for specific problems involving the transfer of energy in dilute gases where the
different densities vary widely it is necessary to solve the heat balance equations directly in order to evaluate the assumption of a single ion temperature which is common to all the ion species present.
III.- RESONANT ION-NEUTRAL COLLISIONS 1. Introduction
Charge exchange, exerts a significant influence upon the momentum transfer cross section and is important for both the problems of ion energy transfer and diffusion. For this process, it is found that although there is a transfer of an electric charge between two particles, each tends to retain its original kinetic energy. Thus, even though the identity of the ion has changed, the reaction is still fundamentally elastic in preserving the total kinetic energy. Charge
1 4 . -
exchange consequently becomes of particular importance to the ion energy balance since it can provide a rapid means for energetic ions to be transformed into energetic neutral particles. For many of the possible atmospheric ion-neutral collision pairs-it is found that the probability for charge exchange is much larger than that for the normal gas kinetic collisions. Therefore, while an original- ly energetic ion could cool to the ambient gas temperature by means of multiple elastic collisions, it can lose all of its excess energy in a single charge exchange reaction.
Charge exchange also plays an important part in .enhancing the magnitude of the momentum transfer cross section above those values normally associated with high temperature ion-neutral reactions. This action arises from the conversion of very slightly glancing collisions into what appear to be nearly direct impacts with a consequent back- scattering. Under these circumstances there exists a relation between the charge exchange, q , and momentum transfer, q , cross sections given
E D by Dalgarno (1958) as q = 2q . This relationship is valid only for
D £
high temperatures where the ion-neutral polarization effects can be neglected.
An adequate introduction to the details of the quantum theory of resonance charge exchange can be found in Bates (1962). In general, an exact solution to the problem of determining the charge exchange cross section is not possible because of the uncertainties in the form of the energy wave functions for all possible energy states of two colliding particles. Therefore, it is necessary to resort to experimental results to determine specific resonance charge exchange cross sections.
2. R e s o n a n c e Gross Sections - T h e o r y
Since w e are interesed in both the charge exchange and m o m e n t u m transfer cross sections it must be noted that to determine
the full temperature variations of the m o m e n t u m transfer cross sec- tion it is not adequate to use the relation q ^ = F o r t e mPe r a~ tures below a transition region, generally in the neighborhood of 500°K, the m o m e n t u m transfer cross section is determined largely by the induced dipole polarization force between the ion and the neutral atom. The appropriate cross section for this interaction
is given by equation (5) as
~ 3 3/2 ( 4 . 8 8 a e2)1 / 2
" (kT„ + kT ) ' l n
For temperatures above the transition region- the polarization contribution to Q ^ diminishes rapidly and the effect of the charge exchange enchancement of the m o m e n t u m transfer cross section becomes dominant. This problem has been discussed by Gosh and Sharma (1964).
It has been shown by Dalgarno (1958) that the velocity dependent charge exchange cross section, q , can be expressed as
E
qE - (A - Blog1 ( )e) , (23) 2
for particle energies below 500 ev. Here A and B are constants
characteristic of a given gas and e is the kinetic energy of relative motion, measured in electron volts (ev). Using q_ = 2q , the high
D E
temperature velocity dependent m o m e n t u m transfer cross section can be given as
qD - 2(A - B In e )2 (24)
16.-
It is important to note that the total cross section for momentum transfer is not equal to the sum of the charge exchange and polarization contributions. Because there exist mutual interaction effects, the polarization potential and charge exchange action combine to form a total cross section whose variation with temperature follows closely the individual cross sections in their own regions of dominance.
In the transition region there is a slight enchancement of the cross section to values above those due to either polarization or charge exchange, but this extends only over a limited temperature range.
Thus, for temperatures above the transition region, q^ is determined exclusively, to within 1% (Knof, et al., 1964), by charge exchange, while below this point the polarization forces are predominant. This effect illustrated in Figure 1 for the specific example oi - N^
collisions. Here the transition region extends , invterms of
r = T.+ T , from 200° to 700°K and is centered about the point r = 340°K.
i n
It is seen that for r> 750°K there is essentially no contribution to Q^ from the polarization.
In order to determine suitable temperature dependent average charge exchange and momentum transfer cross sections, it is necessary to average equations (23) and (24) over the ion and neutral velocity distributions. Thus,
Q_ - 2 Q = ,/Tf.f q d3 v.d3 t (25)
D E
JJ i
n D i nwhere q is given by equation (24), f is the particle velocity distribu-
D 3_i.
tion function, d v is a velocity space volume element, and the subscripts refer to ions and neutrals, respectively.
The solution to equation (25) for conditions where T.=» T x n has been given by Mason and Vanderslice (1959) and Sheldon (1964).
For situations where the ions and neutrals have separate Maxwellian distributions with T. # T it has been possible to generalize these
i n results to obtain
16
8
1 T 1 1 1 1—I—p
N + - N
2MOMENTUM TRANSFER CROSS SECTION POLARIZATION
J I I I I—L
CHARGE EXCHANGE
I I I I I L
100 200 500 1000
T
L +T
N( ° K )
2000 5000 10000
J U
1.- N_ - N. Momentum Transfer Cross Section. The polarization curve is based upon a molecular polarizabi1ity of 1.76 x 10"2* cm3 while the charge exchange contribution is taken from the experimental data of Amine and Utterback (1964) extrapolated to thermal energies.
1 8 . -
Q „ = [(A + 3.96B) - B Log. r ] 2 x 1 0 ~1 6 c m2, (26)
£> iU
~ 8
where A and B , measured in units of 10 c m , are the coefficients appearing in the equation
q£ = (A - B L o g1 ( )£ )2 . (27)
Following equation (2), we now define a resonance ion-neutral collision frequency, v , as
R
4 f 8k\ 1/2 . 1/2 — ,o o x
VR = 3 "n ( V T ) QD ' ( 2 8>
with m the ion or neutral particle m a s s . In a similar manner w e define the average collision frequency for charge exchange, v , as
E
the factor of .4/3 appearing since we desire to use this equation in an energy transfer equation which is analogous to equation (1), the mass factor being omitted* From equations (28) and (29), w e have
for the temperature regime where polarization effects are negligible.
3 . Resonance Ion-Neutral Energy Transfer
The derivation of the energy transfer rate of an ion gas under resonance conditions requires the use of the ion continuity equation with the charge exchange collision frequency. T h u s , in the present section, the general equation of collisional energy loss does not a p p l y .
The energy loss rate of an ion gas follows from consideration of the energy balance of ion production and loss. T h u s ,
1 9 . -
{Eini dU.
IT 3 - 1 EiniVE - EnniVE y ( 3 1 ) where E^ and E^ are the respective ion and neutral particle energies,
U . is total ion gas energy per unit volume, and v is the charge exchange
i E collision frequency defined b y equation (29). If w e apply the relations
E. = 3kT,/2 and E 3kT / 2 , w h i c h apply for gases with Maxwellian
i i n n
velocity distributions, w e obtain
dU. _ i 3
dt 2 nik VE ( Ti " ' ( 3 2 )
or,
d Ui - 4 " -3 -1
— - - 1.-3 x 10 n . (T.- T )v_ ev cm sec . (33) dt 1 x n E
To apply the preceding results to obtain accurate values for the resonance ion-neutral collision frequencies and energy transfer rates it is necessary to know the charge exchange and m o m e n t u m transfer cross sections for each ion. T h u s , in the following section an
analysis is made of the experimental and theoretical results for each atmospheric ion in order to arrive at satisfactory values for Q and Q„,
D b These quantities are then used to obtain adequate expressions for the
appropriate collision frequencies and the ion energy loss rates.
4. Resonance Experimental Cross Sections
A . Atomic Hydrogen
The charge exchange reaction H+ + H H + H+ has been studied extensively because of its theoretical simplicity. Dalgarno and
Yadev (1953) made accurate calculations using perturbed stationary state w a v e functions. Dalgarno (1958), in calculating the mobility of ions in parent gases, w a s able to reproduce essentially the same
20.-
results and c o m p a r e them w i t h e x p e r i m e n t . G u r n e e and M a g e e (1957).
applied the s e m i - c l a s s i c a l impact p a r a m e t e r m e t h o d to obtain v a l u e s of the c h a r g e exchange cross section at two ion v e l o c i t i e s . R a p p and O r t e n b u r g e r (1960) found their t h e o r e t i c a l approach g a v e v a l u e s in
s u b s t a n t i a l a g r e e m e n t w i t h those of D a l g a r n o and Y a d e v . L a t e r , ex- tensions w e r e m a d e b y R a p p and F r a n c i s (1962) .
E x p e r i m e n t a l data have b e e n obtained b y F i t e , et a l . (1960) b y u s i n g b e a m techniques w i t h ion energies as low as 4 0 0 e v . T h e data w e r e later extended b y F i t e , et a l . (1962) to a lower limit of 2 0 e v , giving c l o s e a g r e e m e n t w i t h the p r e v i o u s r e s u l t s . T h e v a l u e s of the v a r i o u s and t h e o r e t i c a l m e a s u r e m e n t s are listed in T a b l e 3 .
F o r the p r e s e n t p u r p o s e s , the c h a r g e e x c h a n g e c r o s s section g i v e n b y D a l g a r n o and Y a d e v (1953) appears to represent the b e s t c o m p r o m i s e between e x p e r i m e n t and t h e o r y . T h u s , u s i n g equation (26)
T a b l e 3 Atomic H y d r o g e n C h a i g e E x c h a n g e
H+ + H -«• H + H+ 1 / 2
qE - A - B L o g1 ( ) €
Source A x 10 B x 10
8 8
D a l g a r n o (1958) 6.93 0.82 G u r n e e and M a g e e 6.78 0 . 9 1
(1937)
R a p p and Francis 6.05 0 . 8 8 (1962)
F i t e , et a l . (1962) 7.60 1.06 w e obtain for the h y d r o g e n ion charge exchange c r o s s section
QE( H+) = [ 1 0 . 2 - 0 . 8 2 L o g1 ( ) R ]2 x 1 0_ 1 6 c m2 , ( 3 4 )
21.-
while the average momentum transfer cross section corresponding to the high temperature limit is
QD( H+) = [14.4 - 1.16 Log1() R ]2 x 1 0 "1 6 c m2. (35) Here, as in the subsequent equations, T =
To determine the extent of the temperature transition region between the polarization and charge exchange dominance of the momentum transfer cross section, equations (35) and (22) have been examined.
It is found that the temperature transition point where the two
asymptotic effects are equal lies near r = 100°K. Thus, for the case of atomic hydrogen, there is essentially no error involved in assuming that the charge exchange contribution to Q^ is dominant at all com-*
.bined temperatures above 100°K.
B. Atomic Oxygen
The symmetrical charge exchange reaction 0+ + 0 -» 0 + 0+
has been studied by D a l g a m o (1958) using a semi-classical appro- ximation for the scattering phase shifts. Because there was consi- derable uncertainty in the approximation used to express the inter- action energy of the ion and atom as a function of the radial sepa- ration, accuracy was claimed only to within a factor of two..
Knof, et al. (1964) have made the most detailed study of the oxygen problem by taking explicit account of twelve possible electronic
+
states of the pseudo-molecule. Their results are listed in Table 4, along with those of others authors.
22.-
T a b l e 4 Atomic Oxygen Charge Exchange
0+ + 0 -» 0 + 0+
1/2 .
qE =» A - B L° g1 0 £
Source A x 10 B x 10 8 8
Knof , et al. (1964) 5.59 0 . 4 8 Dalgarno (1958) 9.38 0.57
(see also Dalgarno ;,1964)
Stebbings, et al. (1964) 5.88 0.57 R a p p and Francis (1962) 6.16 0.76
Experimentally, there is good agreement between the values given by
Knof, et al. and the values obtained by Stebbings, et al. (1964). These latter results extend down to 4 0 ev and, according to Knof, et al.,
the average deviation between the predicted and measured values is
only 5.5%» Therefore, the charge exchange cross section of Knof, et al.
will be used here«
Using equation (26) the average charge exchange cross section is
QE( 0+) ^ [7.47 - 0.475 LogjJflx 1 0 ~1 6 c m2. (36)
The transition region between polarization and charge exchange dominance has been investigated. Figure 2 shows the m o m e n t u m transfer cross section and the extension of the asymptotic polarization and charge exchange contributions. The transition is found to take place near r 53 470°K.
It follows from figure 2 that the m a x i m u m error arising from the omis- sion of the polarization effect is 11% at r = 470°K and diminishes rapidly at higher temperatures. Thus, for the computation of the
m o m e n t u m transfer cross section, it is possible to segment the true curve
T i * T
n( ' K )
F i g . 2.- 0+ - 0 M o m e n t u m Transfer Cross S e c t i o n . The polarization curve is based u p o n an atomic polarizability of 0.77 x l O " ^ c m ^ . The charge exchange region is based u p o n the
theoretical w o r k of K n o f , et a l . (1964) expanded to include the effects of differing ion and neutral gas t e m p e r a t u r e s .
24.-
of Figure 2 into two portions : one for charge exchange and the other polarization. Hence,
T > 470°K
r < 470°K
C . Atomic Nitrogen •
Two derivations of the nitrogen charge transfer cross section have been m a d e . Gurnee and Magee (1957) and K n o f , et al (1964) obtained
the expression
qE( N ^ ) - [5.53 - 0.46 L o g1 ( ) e ]2 x l o '1 6 c m2, (38) . which is accepted for the present calculations.
The average charge exchange cross section is, from equation (26), QE( N+) - [ 7 . 3 3 - 0 . 4 5 5 L o g1 ( ) R ]2 x l o "1 6 c m2. ( 3 9 )
. Applying equations (22) and (26) , it is possible to find the asymptotic charge exchange and polarization contributions to Q ^ . However, because the atomic polarizability of atomic nitrogen is only slightly larger than that for atomic oxygen and because the two exchange cross sections are almost identical, it is not necessary to present the data graphically. For atomic nitrogen it is found that the transition temperature corresponds to r = 550°K and that the error at this point in neglecting the polarization contribution is 10%. T h u s , for the present purposes, Q ^ can be taken as
T > 550°K Q (N+) - [10.3 - 0.645 L o g1 n r ]2 x 1 0 ~1 6 c m2
QD( 0+) - [10.5 - 0.67 L o g1 ( )r ]2 x 1 0_ 1 6c m2
QD( 0+)
1.66 x 10
„1/2
•13
cm ( 3 7 )
25.t
_ 2 . 0 x 1 0
-13r < 550°K QD( N ) = ^ cm . (40)
D. Helium
Like atomic hydrogen, helium has been the object of intensive theoretical and experimental study. A paper by Rapp and Francis (1962) summarizies the results of 21 experiments and 4 theoretical studies.
Table 5 Helium Charge Exchange H+ + H -» H
e e e + H+
e 1/2 . _
qE A " B
L o g1 0 é
Source A X 108 B X 10'
Cramer and Simons (1957) 5.25 0.74
Dalgarno (1958) 5.39 0.62
Rapp and Francis (1962) 4.51 0.68 Gurnee and Magee (1957) 5.02 0.64
Hasted (1951) 5.14 0.55
Chanin and Biondi (1957) 4.93 0.82
Dalgarno (1958) calculated both ion mobility and charge transfer cross sections and compared them with the experimental mo- bility data of Chanin and Biondi (1957). Agreement was found to be within 20% at the higher temperatures. Rapp and Francis (1962) applied
an empirical two state model which gives good accord with the experi- mental data for energies above 200 ev. Cramer and Simons (1957) were
able to measure the charge exchange cross section at energies down
to 16 ev. These data agree well with the earlier results of Hasted (1951) and Dillon, et al (1955). The correspondence is very good between the various authors with a spread of 15% at 1000°K, if the results of Rapp and Francis are disregarded. The data of Cramer and Simons listed in Table 5 are used here as representing the charge exchange cross section.
26.-
The average charge exchange cross section for helium may be taken as
Qg(He+) - [8.17 - 0.74 Log1 0 r]2 x lo"16 cm2. (41) For the momentum transfer cross section it is found that.the transition
temperature lies below r = 200°K and virtually no error is involved in neglecting the polarization effect. Consequently, we take
Q_(He+)-[ll.6 - 1.04 Log.„ r] x lo"16 cm2 . (42)
£ 1U
E. Molecular Oxygen
For the reaction 0* + 0^ -» 0^ + 0* there appear to be no theoretically derived values for the charge exchange cross section.
However, several molecular beam experiments have been conducted with energies below 100 ev giving results which can be extrapolated to thermal energies.
Gosh and Sheridan (1957) obtained cross section data for ion energies below 100 ev which agree well with the later measurements of Stebbings, et al. (1963). The earlier work of Potter (1954) does
not appear to be accurate in comparison, since the cross sections given are a factor of two less than the above authors. Amine and Utter-
back (1964) repeated the experiment of Stebbings, et al. while making a close determination of the dependence of the cross sections upon the degree of ionic excitation. At high electron impact energies there was a significant decrease in the charge transfer cross section, indicating that the resonance behavior was easily destroyed by small energy defects between the excited ions and the neutral gas. Table 6 summarizes the various results.
27.-
For the present calculations the data of Amme and
Utterback (1964) are accepted. T h e charge exchange cross section is
Qe( 02) = [7,51 - 5.4 l o g1 Qr ] x 1 0 ' c m . (43)
The relative importance of the polarization and charge ex- change effects in determining can be seen in Figure 3 where the transition
Table 6 Molecular Oxygen Charge Exchange
° 2 + ° 2 ~*°2 + ° 2
1 / 2 • qE = A - B L o g 1 0e
Source A x 10 B x 10 8 8
Stebbings, et a l . (1963) 5.32 0.65 Amme and Utterback (1964) 5.37 0.54 Potter (1954) 3.82 0.36
température is 1600°K. This relatively high temperature appears to be a consequence of the small charge exchange cross section for mo- lecular oxygen. The maximum error associated w i t h the neglect of the polarization effect at the transition temperature is 10%. T h u s , w e m a y approximate the momentum transfer cross section by
r > 1600°K Qd(°2^ s [10.6 - 0.765 L°S1 0 r ]2 x 10°1 6 c m2
(44) -13
_ 2.7 x 10
r < 1600»K QD( 02) - — — 2 c m .
i 1 1 1 — i — i — r
200 500 1000
T i + T n (°K)
2000 5000 10000
N3 CO
F i g . 3.- C>2 - 02 Momentum Transfer Cross Section. The.polarization curve is based upon a molecular polarizability of 1.98 x 1 0 " ^ cnP. The charge exchange curve is based upon the experi- mental data of Amme and Utterback (1964) extrapolated to thermal energies.
29.=
F . Molecular Nitrogen
Theoretical data |for charge exchange in Nj have been given by Gurnee and Magee (1957) and are listed in Table 7 .
Table 7 Molecular Nitrogen Charge Exchange
N * + N2 N2 q_ = A - B
A *
E L o g
i o
eSource A x 1 08 B x 1 08
Amme and Utterback (1964) 7.36 0.68 Stebbings, et a l . (1963) 6.18 0.43 Gurnee and Magee (1957) 8.28 0.58
Potter (1954) 6.05 1.26 .
A number of experimental studies have been conducted, but the problem of ionic excitation has been responsible for inconsistent values. Stebbings, et a l . (1963) have made extensive measurements of
the charge transfer cross section arid have analyzed many of the pre- vious experimental studies in ain effort to determine the sources of error. Amme and Utterback (1964) have recently been able to show that the work of Stebbings, et a l . leads to cross sections which are too low as a consequence of excited molecular ions which are able to des- troy the resonance behavior of t^e charge exchange process; They repeated the experiment using low energy electrons to produce the ni- trogen ions found a considerable increase in the charge exchange cross section.
Using the results of Amme and Utterback (1964) , the charge exchange cross section at thermal energies is taken as
30.-
• 1 - + 2 "1 6 2
Qe( nJ ) - [10.1 - 0.68 L o g1 0 r ] X 10 cm . (45)
The behavior of the momentum transfer cross section is shown in Figure 1 where a transition temperature of r =f 340°K is indicated.
The maximum error involved in neglecting polarization is found to
be 97o at this point. Accepting this uncertainty, the momentum transfer cross section is
r j> 340°K Qp(N*) - [14.2 - 0.92 Log1 ( )r]2 x l o "1 6 cm? (46)
5. Resonance Ion-Neutral Collision Frequencies
With the derivation of the appropriate cross sections equations (28) and (29) can. now be used to find the charge exchange and momentum transfer collision frequencies. The former are of direct importance to the problem of resonance ion-neutral energy transfer while the latter can be applied to the study of ion diffusion. The results for the charge exchange collision frequencies of the atmospheric ions are listed in Table 8. In order to compare the various values it is convenient, to adopt the approximation that T = 3000°K in the logarithm.
This process leads to errors of less than 10% for 1200 < T < 4000°K.
Table 9 lists these reduced charge exchange collision frequencies and shows that there is an order of magnitude difference between the extre-
m u m values. & • • .
Table 8 Average Ion-Neutral Charge Exchange Collision Frequencies Species v (sec
H , H + +
0 , 0 N , N +
He , He
+
+° r °2 N2,
N2
9.5 x 1 0 "1 3 n(H) T1 / 2 [14.4 - 1.15 Log1 ( ) r ]2
2.3 x 1 0 "1 3 n(0) T1 / 2 [10.5 - 0.61 L o g1 Q v ]2
2.6 x 1 0 "1 3 n(N) r1 / 2 [10.3 - 0 . 6 4 Log1 ( ) r ]2
4.4^x 1 0 "1 3 n(He) f V2 [ll.6 - 1.04 Log1 ( ) r ]2
1.7 x 1 0 "1 3 n ( 02) T1 / 2 [10.6 - 0.76 L o g1 0 r ]2
1.8 x 1 0 "1 3 n ( N2) T1 / 2 [14.2 - 0.96 Log1 ( ) r ]2
31.-
Table 9 Reduced Charge Exchange Collision Frequencies L o g1 0 r L o g1 0 (3000°K)
Species H+- H 0+- 0 N+- N H e+- He 0 ^ - 02
N„ N„
Vg (sec ) 1.6 X 10 2.1 X 10°
2.3 X 10*
4.4 X lo"
1.5 X lo"
2.9 X 10°
° n(H) r1 / 2
n(0) r 1/2 n (N) r 1/2 n(He) r n ( 02) r n ( N2) r
1/2 1/2 1/2
The resonance ion neutral collision frequencies for momentum transfer have been obtained and are listed in Table 10. For values of T below the transition temperature it has been assumed that only the polari- zation effect is present. Hence, a given gas can be characterized by two different functional forms for v . These results indicate significant
R
differences exist between the different ions but that for atoms with similar electronic structures,, such as 0 and N, nearly the same values are found.
Table 10 Average Ion-Neutral Resonance Momentum Transfer Collision Frequencies
Species H+, H 0 » 0
+
vR (sec )
r < 100°K 1.9 X 1 0 "1 2 n ( H ) r1 / 2 (14.4-1.15 Log rj*
-13 1/2 2 r > 470°K 4.7 X 10 n(0)r (10.5-0.67 Log. r)
- 1 0
r < 470°K 8.0 X 10 n(0) N+, N
i/o 2 r > 550°K 5.2 X 10 n(N)r ' (10.3-0.64 L o g ^ D
r < 550°K l.Ö X lu"9 n(N)
H e+, He r > 100°K 8.7 X l o "1 3n ( H e ) r1 / 2 (11.6-1.04 L o g ^ r )2
32.-
~ / -K Species Vr (sec )
, _IO I/O 2
oT, 0„ r > 1600°K 3.4 x 10 n(0_)r ' (10.6 - 0.76 Log. JT)
2' 2 -10 2 1 0
r < 1600°K 9.1 X 10 n(02)
N*, N2 r > 340°K 3.6 x 10~1 3 n ( N2> r1 / 2 (14.2 - 0.96 L o g ^ f )2
6. Resonance Ion-Neutral Energy Transfer
The charge exchange collision frequencies given in Table 8 have been used to derive the ion energy loss rates listed in Table 11. A simplification has again been made, however, by noting that the colli- sion cross sections for resonance charge exchange vary only slightly for the normal range of ionospheric ion and neutral temperatures.
Thus, the average value T = 3000°K has been chosen for the logarithmic tern of the cross section.
Table 11 Resonance Ion-Neutral Energy Transfer Rates , -3 -1:
(ev cm sec )
H dU. ,/ 2
~ = - 1.4 x 10"1* n (H ) n d O f f ^ T) ( T ^ T)
0 dU.
- - 2.1 x 10"1 5 n(0+) n(0)(T.+ T )1 / 2 (T.- T) dt l l N dU
, - = - 2.1 x 10"1 5 n(N+) >(N)(T.+ T )1 / 2 (T.- T) dt
He dU. .. . 1 / 2
- - 4.0 x 10 3 n(He ) n(He)(T.+ T) ' (T^-T)
0- dU. , _ , i
™ s - 1.4 x l o " ^ n(02) n(02)(T.+ T)' (Tj_- T)
N_ dU„ i c + 1/2
d T 53 ~ 2 , 7 x 1 0 n ( V n<N2) ( Ti+ T ) ( Ti" T )
33.-
These results can be compared w i t h values adopted in several studies of charged particle temperatures. Hanson (1963) has used an oxygen ion energy loss rate of
d U ( 0 +' 0 ) -13 + - 3 - 1
— — =-1.82 x 10 n ( 0 ) n ( 0 ) (T - T ) ev cm sec , dt i n
(47) based upon i the collision frequencies of Dalgarno (1961b) . A n a l y s i s of this equation indicates that it w a s derived for a temperature of 300°K and, further, that it w a s based upon the m o m e n t u m transfer cross section rather than the charge exchange cross section. How- ever, because Dalgarno®s m o m e n t u m transfer cross section is too large by a factor of 2.7 - 3 . 0 (Knof, et al. 1964), the final result is
larger than the present value by the factors 2.2 - 1.4 over the tem- perature range 1600 < T < 3600°R.
Another calculation of the oxygen ion cooling rate has been m a d e by Dalgarno (1963) . He finds
d V 0 + - - 1 4 + - 3 - 1
— = - 4.47 x 10 n ( 0 ) n ( 0 )(T.- T ) ev cm J sec .
1 (48)
Comparing this equation with that obtained from Table 11, it is seen that the present energy transfer rate is larger by the factor
-2 1/2
4 . 7 x 1 0 T w h i c h , for te varies between 1.9 - 2.8.
- 2 1/2
4 . 7 x 1 0 T w h i c h , for temperatures in the range 1600° < T < 3600°K,
W i l l m o r e (1964) has calculated the effect of h e l i u m reso- nance charge exchange upon the cooling of h e l i u m ions. His result is
dU (He+- H e ) -13 + -3
— = - 1.19 x 10 n(He ) n(He)(TjL~ T ) ev cm J sec (49).
34.-
From Table 11» the corresponding equation from the present data is
- 2
larger than equation (49) by the factor 3.4 x 10 T . For tempera- tures between 1600°K < T < 3600°K, this factor varies between 1.4 and 2 . 0 . At bigher temperatures an increasing divergence # i l l be n o t e d .
Further calculations of resonance ion-neutral energy trans- fer rates are not known to have been m a d e .
IV.- SUMMARY
The primary purpose of this paper has been to present a consistent development of the problems of ion energy transfer and collision frequencies. Knowledge of these factors is needed for any detailed study of the thermal behavior of ions in a dilute, par- tially ionized plasma. In the first section emphasis was placed upon ion-neutral and ion-ion collisions where resonance charge exchange is not an important factor. Using a general equation of energy transfer, adequate expressions for collision frequencies and energy transfer rateswere obtained. For ion-neutral collisions it was necessary to use a momentum transfer cross section appropriate
to an ion-neutral polarization interaction. T h u s , while the present results are probably accurate for temperatures below 300°K, the true behavior at high temperatures must remain in doubt until adequate experimental studies have been m a d e .
The ion-ion collision frequency and energy transfer rate have been computed directly from the coulomb potential»1 llnder conditions of equivalent particle densities the ion-ion energy transfer rate is found to be an order of magnitude larger than the listed ion-neutral rates.
The problem of resonance collisions between an ion and its parent neutral particle has been discussed. Appropriate equations have
35.-
been developed for the thermal nonequilibrium collision frequencies for both charge exchange and momentum trsnsfer. It was shown that ion energy transfer rates for charge exchange must dominate at tem- peratures above the polarization-charge exchange transition region.
For all of the ions, with the exception of » there is little error involved in ignoring the polarization contribution to Q^ in the atmosphere.
In general, both the nonresonant and resonant ion-neutral energy transfer rates show only a small dependence upon the gas temperatures. The resonance behavior leads to rates which are somewhat larger than those found for elastic collisions.
Since the ion-neutral energy transfer rates are generally much larger than electron-neutral and, in a similar fashion, electron ion energy transfer is much slower than ion-ion energy exchange, for an external source giving heat to electrons in a weakly ionized plas- ma there will be a complex set of relations needed to arrive at
equilibrium values of the electron, ion and neutral gas temperatures.
ACKNOWLEDGEMENTS
The guidence of Dr. M. Nicolet is gratefully acknowledged.
This study has been supported, in part, under grant ONR(G) - 00009-66 from the Office of Naval Research. While at the Ionosphere Research Laboratory, the Pennsylvania State University, the author held a NASA Predoctoral Traineeship.
3 6 . »
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